Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A

Percentage Accurate: 69.0% → 99.8%

Time: 12.8s

Alternatives: 17

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x + y} \cdot \frac{\frac{y}{y + \left(x + 1\right)}}{x + y} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (/ x (+ x y)) (/ (/ y (+ y (+ x 1.0))) (+ x y))))

C

double code(double x, double y) {
	return (x / (x + y)) * ((y / (y + (x + 1.0))) / (x + y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / (x + y)) * ((y / (y + (x + 1.0d0))) / (x + y))
end function

Java

public static double code(double x, double y) {
	return (x / (x + y)) * ((y / (y + (x + 1.0))) / (x + y));
}

Python

def code(x, y):
	return (x / (x + y)) * ((y / (y + (x + 1.0))) / (x + y))

Julia

function code(x, y)
	return Float64(Float64(x / Float64(x + y)) * Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(x + y)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x / (x + y)) * ((y / (y + (x + 1.0))) / (x + y));
end

Wolfram

code[x_, y_] := N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.9%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   5. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   7. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

   11. lower-/.f64 99.8

       \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1}}{x + y} \]

   14. associate-+l+ N/A

       \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}}}{x + y} \]

   15. +-commutative N/A

       \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \]

   16. associate-+l+ N/A

       \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

   17. lower-+.f64 N/A

       \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

   18. lower-+.f64 99.8

       \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}}}{x + y} \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{y + \left(1 + x\right)}}{x + y}} \]

5. Final simplification 99.8%

   \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{y + \left(x + 1\right)}}{x + y} \]
6. Add Preprocessing

Alternative 2: 68.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) \cdot \left(x + y\right)\\ \mathbf{if}\;y \leq 5.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 3000000000000:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+160}:\\ \;\;\;\;1 \cdot \frac{x}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ x y) (+ x y))))
   (if (<= y 5.2e-156)
     (* (/ y (+ y (+ x 1.0))) (/ 1.0 x))
     (if (<= y 3000000000000.0)
       (* x (/ y (* t_0 (+ x 1.0))))
       (if (<= y 1.32e+160)
         (* 1.0 (/ x t_0))
         (* (/ x (+ x y)) (/ 1.0 (+ x y))))))))

C

double code(double x, double y) {
	double t_0 = (x + y) * (x + y);
	double tmp;
	if (y <= 5.2e-156) {
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	} else if (y <= 3000000000000.0) {
		tmp = x * (y / (t_0 * (x + 1.0)));
	} else if (y <= 1.32e+160) {
		tmp = 1.0 * (x / t_0);
	} else {
		tmp = (x / (x + y)) * (1.0 / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) * (x + y)
    if (y <= 5.2d-156) then
        tmp = (y / (y + (x + 1.0d0))) * (1.0d0 / x)
    else if (y <= 3000000000000.0d0) then
        tmp = x * (y / (t_0 * (x + 1.0d0)))
    else if (y <= 1.32d+160) then
        tmp = 1.0d0 * (x / t_0)
    else
        tmp = (x / (x + y)) * (1.0d0 / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) * (x + y);
	double tmp;
	if (y <= 5.2e-156) {
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	} else if (y <= 3000000000000.0) {
		tmp = x * (y / (t_0 * (x + 1.0)));
	} else if (y <= 1.32e+160) {
		tmp = 1.0 * (x / t_0);
	} else {
		tmp = (x / (x + y)) * (1.0 / (x + y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) * (x + y)
	tmp = 0
	if y <= 5.2e-156:
		tmp = (y / (y + (x + 1.0))) * (1.0 / x)
	elif y <= 3000000000000.0:
		tmp = x * (y / (t_0 * (x + 1.0)))
	elif y <= 1.32e+160:
		tmp = 1.0 * (x / t_0)
	else:
		tmp = (x / (x + y)) * (1.0 / (x + y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) * Float64(x + y))
	tmp = 0.0
	if (y <= 5.2e-156)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) * Float64(1.0 / x));
	elseif (y <= 3000000000000.0)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(x + 1.0))));
	elseif (y <= 1.32e+160)
		tmp = Float64(1.0 * Float64(x / t_0));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) * (x + y);
	tmp = 0.0;
	if (y <= 5.2e-156)
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	elseif (y <= 3000000000000.0)
		tmp = x * (y / (t_0 * (x + 1.0)));
	elseif (y <= 1.32e+160)
		tmp = 1.0 * (x / t_0);
	else
		tmp = (x / (x + y)) * (1.0 / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.2e-156], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3000000000000.0], N[(x * N[(y / N[(t$95$0 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.32e+160], N[(1.0 * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 5.2000000000000002e-156

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      10. associate-+l+ N/A

          \[\leadsto \frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      15. lower-/.f64 86.6

          \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   4. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 58.2

         \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{x}} \]

   7. Applied rewrites 58.2%

      \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{x}} \]

   if 5.2000000000000002e-156 < y < 3e12

   1. Initial program 85.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      6. lower-/.f64 96.1

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

      9. associate-+l+ N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \cdot x \]

      10. +-commutative N/A

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \cdot x \]

      11. associate-+l+ N/A

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \cdot x \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \cdot x \]

      13. lower-+.f64 96.1

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + \color{blue}{\left(1 + x\right)}\right)} \cdot x \]

   4. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + \left(1 + x\right)\right)} \cdot x} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + 1\right)}} \cdot x \]

      2. lower-+.f64 93.2

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + 1\right)}} \cdot x \]

   7. Applied rewrites 93.2%

      \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + 1\right)}} \cdot x \]

   if 3e12 < y < 1.32e160

   1. Initial program 59.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      10. associate-+l+ N/A

          \[\leadsto \frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      15. lower-/.f64 90.9

          \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.3%

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

   if 1.32e160 < y

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      11. lower-/.f64 100.0

          \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      18. lower-+.f64 100.0

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}}}{x + y} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{y + \left(1 + x\right)}}{x + y}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{1}}{x + y} \]
   6. Step-by-step derivation

   7. Applied rewrites 88.2%

      \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{1}}{x + y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 3000000000000:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+160}:\\ \;\;\;\;1 \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 5.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{t\_0} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y 5.2e-156)
     (* (/ y t_0) (/ 1.0 x))
     (if (<= y 4.1e+96)
       (* x (/ y (* t_0 (* (+ x y) (+ x y)))))
       (* (/ x (+ x y)) (/ 1.0 (+ x y)))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 5.2e-156) {
		tmp = (y / t_0) * (1.0 / x);
	} else if (y <= 4.1e+96) {
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) * (1.0 / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= 5.2d-156) then
        tmp = (y / t_0) * (1.0d0 / x)
    else if (y <= 4.1d+96) then
        tmp = x * (y / (t_0 * ((x + y) * (x + y))))
    else
        tmp = (x / (x + y)) * (1.0d0 / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 5.2e-156) {
		tmp = (y / t_0) * (1.0 / x);
	} else if (y <= 4.1e+96) {
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) * (1.0 / (x + y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 5.2e-156:
		tmp = (y / t_0) * (1.0 / x)
	elif y <= 4.1e+96:
		tmp = x * (y / (t_0 * ((x + y) * (x + y))))
	else:
		tmp = (x / (x + y)) * (1.0 / (x + y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= 5.2e-156)
		tmp = Float64(Float64(y / t_0) * Float64(1.0 / x));
	elseif (y <= 4.1e+96)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= 5.2e-156)
		tmp = (y / t_0) * (1.0 / x);
	elseif (y <= 4.1e+96)
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	else
		tmp = (x / (x + y)) * (1.0 / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.2e-156], N[(N[(y / t$95$0), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e+96], N[(x * N[(y / N[(t$95$0 * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.2000000000000002e-156

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      10. associate-+l+ N/A

          \[\leadsto \frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      15. lower-/.f64 86.6

          \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   4. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 58.2

         \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{x}} \]

   7. Applied rewrites 58.2%

      \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{x}} \]

   if 5.2000000000000002e-156 < y < 4.09999999999999998e96

   1. Initial program 81.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      6. lower-/.f64 92.3

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

      9. associate-+l+ N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \cdot x \]

      10. +-commutative N/A

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \cdot x \]

      11. associate-+l+ N/A

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \cdot x \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \cdot x \]

      13. lower-+.f64 92.3

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + \color{blue}{\left(1 + x\right)}\right)} \cdot x \]

   4. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + \left(1 + x\right)\right)} \cdot x} \]

   if 4.09999999999999998e96 < y

   1. Initial program 46.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      11. lower-/.f64 99.9

          \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      18. lower-+.f64 99.9

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}}}{x + y} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{y + \left(1 + x\right)}}{x + y}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{1}}{x + y} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.5%

      \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{1}}{x + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + \left(x + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 1.32 \cdot 10^{+160}:\\ \;\;\;\;\frac{y \cdot t\_0}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y 1.32e+160)
     (/ (* y t_0) (* (+ x y) (+ y (+ x 1.0))))
     (* t_0 (/ 1.0 (+ x y))))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 1.32e+160) {
		tmp = (y * t_0) / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = t_0 * (1.0 / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (y <= 1.32d+160) then
        tmp = (y * t_0) / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = t_0 * (1.0d0 / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 1.32e+160) {
		tmp = (y * t_0) / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = t_0 * (1.0 / (x + y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= 1.32e+160:
		tmp = (y * t_0) / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = t_0 * (1.0 / (x + y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= 1.32e+160)
		tmp = Float64(Float64(y * t_0) / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= 1.32e+160)
		tmp = (y * t_0) / ((x + y) * (y + (x + 1.0)));
	else
		tmp = t_0 * (1.0 / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.32e+160], N[(N[(y * t$95$0), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.32e160

   1. Initial program 69.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y} \cdot y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}} \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-*.f64 95.9

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

      18. lower-+.f64 95.9

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(y + \color{blue}{\left(1 + x\right)}\right)} \]

   4. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

   if 1.32e160 < y

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      11. lower-/.f64 100.0

          \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      18. lower-+.f64 100.0

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}}}{x + y} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{y + \left(1 + x\right)}}{x + y}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{1}}{x + y} \]
   6. Step-by-step derivation

   7. Applied rewrites 88.2%

      \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{1}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{+160}:\\ \;\;\;\;\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 1.32 \cdot 10^{+160}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y 1.32e+160)
     (* t_0 (/ y (* (+ x y) (+ y (+ x 1.0)))))
     (* t_0 (/ 1.0 (+ x y))))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 1.32e+160) {
		tmp = t_0 * (y / ((x + y) * (y + (x + 1.0))));
	} else {
		tmp = t_0 * (1.0 / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (y <= 1.32d+160) then
        tmp = t_0 * (y / ((x + y) * (y + (x + 1.0d0))))
    else
        tmp = t_0 * (1.0d0 / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 1.32e+160) {
		tmp = t_0 * (y / ((x + y) * (y + (x + 1.0))));
	} else {
		tmp = t_0 * (1.0 / (x + y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= 1.32e+160:
		tmp = t_0 * (y / ((x + y) * (y + (x + 1.0))))
	else:
		tmp = t_0 * (1.0 / (x + y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= 1.32e+160)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0)))));
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= 1.32e+160)
		tmp = t_0 * (y / ((x + y) * (y + (x + 1.0))));
	else
		tmp = t_0 * (1.0 / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.32e+160], N[(t$95$0 * N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.32e160

   1. Initial program 69.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right) \cdot \left(x + y\right)}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot \frac{x}{x + y}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot \frac{x}{x + y}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot \frac{x}{x + y} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot \frac{x}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot \frac{x}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot \frac{x}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \cdot \frac{x}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \cdot \frac{x}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \cdot \frac{x}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \cdot \frac{x}{x + y} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(y + \color{blue}{\left(1 + x\right)}\right)} \cdot \frac{x}{x + y} \]

      19. lower-/.f64 95.9

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(y + \left(1 + x\right)\right)} \cdot \color{blue}{\frac{x}{x + y}} \]

   4. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(y + \left(1 + x\right)\right)} \cdot \frac{x}{x + y}} \]

   if 1.32e160 < y

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      11. lower-/.f64 100.0

          \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      18. lower-+.f64 100.0

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}}}{x + y} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{y + \left(1 + x\right)}}{x + y}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{1}}{x + y} \]
   6. Step-by-step derivation

   7. Applied rewrites 88.2%

      \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{1}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{+160}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-201}:\\ \;\;\;\;\frac{y \cdot 1}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+154)
   (/ (/ y x) x)
   (if (<= x -4.7e-201)
     (/ (* y 1.0) (* (+ x y) (+ y (+ x 1.0))))
     (if (<= x 3.7e+24) (/ x (fma y y y)) (* (/ x (+ x y)) (/ 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+154) {
		tmp = (y / x) / x;
	} else if (x <= -4.7e-201) {
		tmp = (y * 1.0) / ((x + y) * (y + (x + 1.0)));
	} else if (x <= 3.7e+24) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / (x + y)) * (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+154)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -4.7e-201)
		tmp = Float64(Float64(y * 1.0) / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	elseif (x <= 3.7e+24)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e+154], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -4.7e-201], N[(N[(y * 1.0), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.7e+24], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.4e154

   1. Initial program 52.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 78.4

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.1%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if -1.4e154 < x < -4.69999999999999994e-201

   1. Initial program 85.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y} \cdot y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}} \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-*.f64 98.6

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

      18. lower-+.f64 98.6

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(y + \color{blue}{\left(1 + x\right)}\right)} \]

   4. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{1} \cdot y}{\left(x + y\right) \cdot \left(y + \left(1 + x\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.1%

      \[\leadsto \frac{\color{blue}{1} \cdot y}{\left(x + y\right) \cdot \left(y + \left(1 + x\right)\right)} \]

   if -4.69999999999999994e-201 < x < 3.69999999999999999e24

   1. Initial program 67.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 80.3

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 80.3%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 3.69999999999999999e24 < x

   1. Initial program 53.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      11. lower-/.f64 99.7

          \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      18. lower-+.f64 99.7

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}}}{x + y} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{y + \left(1 + x\right)}}{x + y}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{y}} \]
   6. Step-by-step derivation

      1. lower-/.f64 31.2

         \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{y}} \]

   7. Applied rewrites 31.2%

      \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-201}:\\ \;\;\;\;\frac{y \cdot 1}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+160}:\\ \;\;\;\;1 \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.5e-139)
   (/ y (fma x x x))
   (if (<= y 4.3e+14)
     (/ x (fma y y y))
     (if (<= y 1.32e+160)
       (* 1.0 (/ x (* (+ x y) (+ x y))))
       (* (/ x (+ x y)) (/ 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.5e-139) {
		tmp = y / fma(x, x, x);
	} else if (y <= 4.3e+14) {
		tmp = x / fma(y, y, y);
	} else if (y <= 1.32e+160) {
		tmp = 1.0 * (x / ((x + y) * (x + y)));
	} else {
		tmp = (x / (x + y)) * (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.5e-139)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 4.3e+14)
		tmp = Float64(x / fma(y, y, y));
	elseif (y <= 1.32e+160)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.5e-139], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.3e+14], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.32e+160], N[(1.0 * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 7.5000000000000001e-139

   1. Initial program 67.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 58.2

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 7.5000000000000001e-139 < y < 4.3e14

   1. Initial program 85.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 47.1

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 4.3e14 < y < 1.32e160

   1. Initial program 59.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      10. associate-+l+ N/A

          \[\leadsto \frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      15. lower-/.f64 90.9

          \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.3%

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

   if 1.32e160 < y

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      11. lower-/.f64 100.0

          \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      18. lower-+.f64 100.0

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}}}{x + y} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{y + \left(1 + x\right)}}{x + y}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{y}} \]
   6. Step-by-step derivation

      1. lower-/.f64 87.9

         \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{y}} \]

   7. Applied rewrites 87.9%

      \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 61.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+160}:\\ \;\;\;\;1 \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.5e-139)
   (/ y (fma x x x))
   (if (<= y 4.3e+14)
     (/ x (fma y y y))
     (if (<= y 1.32e+160)
       (* 1.0 (/ x (* (+ x y) (+ x y))))
       (* (/ 1.0 y) (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.5e-139) {
		tmp = y / fma(x, x, x);
	} else if (y <= 4.3e+14) {
		tmp = x / fma(y, y, y);
	} else if (y <= 1.32e+160) {
		tmp = 1.0 * (x / ((x + y) * (x + y)));
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.5e-139)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 4.3e+14)
		tmp = Float64(x / fma(y, y, y));
	elseif (y <= 1.32e+160)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.5e-139], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.3e+14], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.32e+160], N[(1.0 * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 7.5000000000000001e-139

   1. Initial program 67.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 58.2

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 7.5000000000000001e-139 < y < 4.3e14

   1. Initial program 85.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 47.1

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 4.3e14 < y < 1.32e160

   1. Initial program 59.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      10. associate-+l+ N/A

          \[\leadsto \frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      15. lower-/.f64 90.9

          \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.3%

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

   if 1.32e160 < y

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. lower-*.f64 71.2

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.7%

      \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{1}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+160}:\\ \;\;\;\;1 \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{t\_0} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-201}:\\ \;\;\;\;\frac{y \cdot 1}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.4e+154)
     (* (/ y t_0) (/ 1.0 x))
     (if (<= x -4.7e-201)
       (/ (* y 1.0) (* (+ x y) t_0))
       (* (/ x (+ x y)) (/ 1.0 (+ y 1.0)))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.4e+154) {
		tmp = (y / t_0) * (1.0 / x);
	} else if (x <= -4.7e-201) {
		tmp = (y * 1.0) / ((x + y) * t_0);
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.4d+154)) then
        tmp = (y / t_0) * (1.0d0 / x)
    else if (x <= (-4.7d-201)) then
        tmp = (y * 1.0d0) / ((x + y) * t_0)
    else
        tmp = (x / (x + y)) * (1.0d0 / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.4e+154) {
		tmp = (y / t_0) * (1.0 / x);
	} else if (x <= -4.7e-201) {
		tmp = (y * 1.0) / ((x + y) * t_0);
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.4e+154:
		tmp = (y / t_0) * (1.0 / x)
	elif x <= -4.7e-201:
		tmp = (y * 1.0) / ((x + y) * t_0)
	else:
		tmp = (x / (x + y)) * (1.0 / (y + 1.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.4e+154)
		tmp = Float64(Float64(y / t_0) * Float64(1.0 / x));
	elseif (x <= -4.7e-201)
		tmp = Float64(Float64(y * 1.0) / Float64(Float64(x + y) * t_0));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.4e+154)
		tmp = (y / t_0) * (1.0 / x);
	elseif (x <= -4.7e-201)
		tmp = (y * 1.0) / ((x + y) * t_0);
	else
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e+154], N[(N[(y / t$95$0), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.7e-201], N[(N[(y * 1.0), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e154

   1. Initial program 52.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      10. associate-+l+ N/A

          \[\leadsto \frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      15. lower-/.f64 78.4

          \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   4. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 87.4

         \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{x}} \]

   7. Applied rewrites 87.4%

      \[\leadsto \frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{x}} \]

   if -1.4e154 < x < -4.69999999999999994e-201

   1. Initial program 85.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y} \cdot y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}} \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-*.f64 98.6

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

      18. lower-+.f64 98.6

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(y + \color{blue}{\left(1 + x\right)}\right)} \]

   4. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{1} \cdot y}{\left(x + y\right) \cdot \left(y + \left(1 + x\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.1%

      \[\leadsto \frac{\color{blue}{1} \cdot y}{\left(x + y\right) \cdot \left(y + \left(1 + x\right)\right)} \]

   if -4.69999999999999994e-201 < x

   1. Initial program 61.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      11. lower-/.f64 99.8

          \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      18. lower-+.f64 99.8

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}}}{x + y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{y + \left(1 + x\right)}}{x + y}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{1 + y}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{1 + y}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{x + y} \cdot \frac{1}{\color{blue}{y + 1}} \]

      3. lower-+.f64 61.6

         \[\leadsto \frac{x}{x + y} \cdot \frac{1}{\color{blue}{y + 1}} \]

   7. Applied rewrites 61.6%

      \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{y + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-201}:\\ \;\;\;\;\frac{y \cdot 1}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-201}:\\ \;\;\;\;\frac{y \cdot 1}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+154)
   (/ (/ y x) x)
   (if (<= x -4.7e-201)
     (/ (* y 1.0) (* (+ x y) (+ y (+ x 1.0))))
     (* (/ x (+ x y)) (/ 1.0 (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+154) {
		tmp = (y / x) / x;
	} else if (x <= -4.7e-201) {
		tmp = (y * 1.0) / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.4d+154)) then
        tmp = (y / x) / x
    else if (x <= (-4.7d-201)) then
        tmp = (y * 1.0d0) / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = (x / (x + y)) * (1.0d0 / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+154) {
		tmp = (y / x) / x;
	} else if (x <= -4.7e-201) {
		tmp = (y * 1.0) / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4e+154:
		tmp = (y / x) / x
	elif x <= -4.7e-201:
		tmp = (y * 1.0) / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = (x / (x + y)) * (1.0 / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+154)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -4.7e-201)
		tmp = Float64(Float64(y * 1.0) / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.4e+154)
		tmp = (y / x) / x;
	elseif (x <= -4.7e-201)
		tmp = (y * 1.0) / ((x + y) * (y + (x + 1.0)));
	else
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e+154], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -4.7e-201], N[(N[(y * 1.0), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e154

   1. Initial program 52.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 78.4

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.1%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if -1.4e154 < x < -4.69999999999999994e-201

   1. Initial program 85.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y} \cdot y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}} \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-*.f64 98.6

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

      18. lower-+.f64 98.6

          \[\leadsto \frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(y + \color{blue}{\left(1 + x\right)}\right)} \]

   4. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot y}{\left(x + y\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{1} \cdot y}{\left(x + y\right) \cdot \left(y + \left(1 + x\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.1%

      \[\leadsto \frac{\color{blue}{1} \cdot y}{\left(x + y\right) \cdot \left(y + \left(1 + x\right)\right)} \]

   if -4.69999999999999994e-201 < x

   1. Initial program 61.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{y}{\left(x + y\right) + 1}}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      11. lower-/.f64 99.8

          \[\leadsto \frac{x}{x + y} \cdot \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      18. lower-+.f64 99.8

          \[\leadsto \frac{x}{x + y} \cdot \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}}}{x + y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{y + \left(1 + x\right)}}{x + y}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{1 + y}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{1 + y}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{x + y} \cdot \frac{1}{\color{blue}{y + 1}} \]

      3. lower-+.f64 61.6

         \[\leadsto \frac{x}{x + y} \cdot \frac{1}{\color{blue}{y + 1}} \]

   7. Applied rewrites 61.6%

      \[\leadsto \frac{x}{x + y} \cdot \color{blue}{\frac{1}{y + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-201}:\\ \;\;\;\;\frac{y \cdot 1}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+160}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.5e-139)
   (/ y (fma x x x))
   (if (<= y 1.32e+160) (/ x (fma y y y)) (* (/ 1.0 y) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.5e-139) {
		tmp = y / fma(x, x, x);
	} else if (y <= 1.32e+160) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.5e-139)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 1.32e+160)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.5e-139], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.32e+160], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.5000000000000001e-139

   1. Initial program 67.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 58.2

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 7.5000000000000001e-139 < y < 1.32e160

   1. Initial program 73.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 52.2

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 1.32e160 < y

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. lower-*.f64 71.2

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.7%

      \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{1}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+160}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-185}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= x -9.2e+16)
     (/ y (* x x))
     (if (<= x -5.3e-89) t_0 (if (<= x 1.9e-185) (/ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -9.2e+16) {
		tmp = y / (x * x);
	} else if (x <= -5.3e-89) {
		tmp = t_0;
	} else if (x <= 1.9e-185) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (x <= (-9.2d+16)) then
        tmp = y / (x * x)
    else if (x <= (-5.3d-89)) then
        tmp = t_0
    else if (x <= 1.9d-185) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -9.2e+16) {
		tmp = y / (x * x);
	} else if (x <= -5.3e-89) {
		tmp = t_0;
	} else if (x <= 1.9e-185) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if x <= -9.2e+16:
		tmp = y / (x * x)
	elif x <= -5.3e-89:
		tmp = t_0
	elif x <= 1.9e-185:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (x <= -9.2e+16)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -5.3e-89)
		tmp = t_0;
	elseif (x <= 1.9e-185)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (x <= -9.2e+16)
		tmp = y / (x * x);
	elseif (x <= -5.3e-89)
		tmp = t_0;
	elseif (x <= 1.9e-185)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.2e+16], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.3e-89], t$95$0, If[LessEqual[x, 1.9e-185], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.2e16

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 78.2

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -9.2e16 < x < -5.3000000000000001e-89 or 1.9e-185 < x

   1. Initial program 68.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. lower-*.f64 36.6

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]

   if -5.3000000000000001e-89 < x < 1.9e-185

   1. Initial program 63.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 87.8

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.6%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 59.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+160}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.5e-139)
   (/ y (fma x x x))
   (if (<= y 1.32e+160) (/ x (fma y y y)) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.5e-139) {
		tmp = y / fma(x, x, x);
	} else if (y <= 1.32e+160) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.5e-139)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 1.32e+160)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.5e-139], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.32e+160], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.5000000000000001e-139

   1. Initial program 67.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 58.2

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 7.5000000000000001e-139 < y < 1.32e160

   1. Initial program 73.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 52.2

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 1.32e160 < y

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. lower-*.f64 71.2

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.7%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 59.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.5e-139) (/ y (fma x x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.5e-139) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.5e-139)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.5e-139], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5000000000000001e-139

   1. Initial program 67.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 58.2

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 7.5000000000000001e-139 < y

   1. Initial program 65.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 58.2

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 61.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.2e+16) (/ y (* x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.2e+16) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.2e+16)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.2e+16], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.2e16

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 78.2

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -9.2e16 < x

   1. Initial program 66.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 62.3

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 36.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.0) (/ x y) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 70.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 44.4

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.2%

      \[\leadsto \frac{x}{\color{blue}{y}} \]

   if 1 < y

   1. Initial program 55.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. lower-*.f64 63.1

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 26.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x y))

C

double code(double x, double y) {
	return x / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

public static double code(double x, double y) {
	return x / y;
}

Python

def code(x, y):
	return x / y

Julia

function code(x, y)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 66.9%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

   4. *-rgt-identity N/A

      \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

   5. lower-fma.f64 49.3

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

5. Applied rewrites 49.3%

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{x}{\color{blue}{y}} \]
7. Step-by-step derivation

8. Applied rewrites 30.4%

   \[\leadsto \frac{x}{\color{blue}{y}} \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024222 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))